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Introduction to applied mathematics. (English) Zbl 0618.00015
Wellesley, Massachusetts 02181: Wellesley-Cambridge Press. XII, 758 p. (1986).
The appearance of a book with this title is not likely to cause much comment. However the present book is unusual and contains much which would not be expected. It opens with a chapter entitled symmetric linear systems. As well as the matters which might be expected to be discussed in this, there are discussions of the solution of problems defined by the matrix equation \(Ax=b\) by the use of least squares and of normal modes. The second chapter, equilibrium equations and minimum principles, continues the work of the first and introduces topics such as dual problems, Lagrange multipliers and the Kalman filter.
The third chapter, equilibrium in the continuous case, deals with a variety of matters. The calculus of variations and complementary minimum principles are discussed and the idea of the spline is introduced. There is also a short treatment of the derivation of a number of partial differential equations of Mathematical Physics. The fourth chapter, analytical methods, gives a fairly conventional treatment of subjects such as complex variables, Fourier Series and Integrals and Laplace Transforms.
The fifth chapter, on numerical methods, begins with a discussion of various methods of solving the problems represented by the matrix equations \(Ax=b\) and \(AX=\lambda x\). What adds additional interest is the inclusion here of a treatment of the finite element method and the fast Fourier Transform, subjects which have previously received little attention in general textbooks. In the sixth chapter, entitled initial value problems, the author discusses a number of fairly conventional topics such as ordinary differential equations, the z transform, the heat and wave equations and their numerical solution. In addition however there are two interesting topics which have become of importance recently - chaos and solitons.
The two final chapters, network flows and combinatorics, and optimization deal with various aspects of operational research, such as the transportation problem, and linear and nonlinear programming. In some ways one feels that these two chapters do not really belong to the book, but have been put in to make certain that the reader is aware that there are domains of Applied Mathematics outwith the physical sciences. It would perhaps have been better to have instead included a treatment of Groups and special functions and more extensive treatments of topics such as Integral Equations, Green’s functions and Hilbert spaces. The book closes with information about sources of relevant software and references.
This is a fascinating book. One is amazed that one person could write all this. The exposition is clear and the author brings life to what is often regarded as a bread and butter subject. The book should be on the shelves of everyone teaching Applied Mathematics and mathematical methods. It will also be of great interest to students who realise that there is more to mathematics than passing examinations. The printing and presentation are excellent and the book is thoroughly recommended.
Reviewer: Ll.G.Chambers

MSC:
00A69 General applied mathematics
90-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to operations research and mathematical programming